## The Calabi invariant and the Euler class

HTML articles powered by AMS MathViewer

- by Takashi Tsuboi PDF
- Trans. Amer. Math. Soc.
**352**(2000), 515-524 Request permission

## Abstract:

We show the following relationship between the Euler class for the group of the orientation preserving diffeomorphisms of the circle and the Calabi invariant for the group of area preserving diffeomorphisms of the disk which are the identity along the boundary. A diffeomorphism of the circle admits an extension which is an area preserving diffeomorphism of the disk. For a homomorphism $\psi$ from the fundamental group $\langle a_{1}, \cdots , a_{2g} ; [a_{1},a_{2}]\cdots [a_{2g-1},a_{2g}]\rangle$ of a closed surface to the group of the diffeomorphisms of the circle, by taking the extensions $\widetilde {\psi (a}_{i})$ for the generators $a_{i}$, one obtains the product $[\widetilde {\psi (a}_{1}),\widetilde {\psi (a}_{2})]\cdots [\widetilde {\psi (a}_{2g-1}),\widetilde {\psi (a}_{2g})]$ of their commutators, and this is an area preserving diffeomorphism of the disk which is the identity along the boundary. Then the Calabi invariant of this area preserving diffeomorphism is a non-zero multiple of the Euler class of the associated circle bundle evaluated on the fundamental cycle of the surface.## References

- Augustin Banyaga,
*Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique*, Comment. Math. Helv.**53**(1978), no. 2, 174–227 (French). MR**490874**, DOI 10.1007/BF02566074 - Eugenio Calabi,
*On the group of automorphisms of a symplectic manifold*, Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 1–26. MR**0350776** - A. Fathi,
*Structure of the group of homeomorphisms preserving a good measure on a compact manifold*, Ann. Sci. École Norm. Sup. (4)**13**(1980), no. 1, 45–93. MR**584082** - Jean-Marc Gambaudo and Étienne Ghys,
*Enlacements asymptotiques*, Topology**36**(1997), no. 6, 1355–1379 (French). MR**1452855**, DOI 10.1016/S0040-9383(97)00001-3 - Peter Greenberg,
*Generators and relations in the classifying space for pl foliations*, Topology Appl.**48**(1992), no. 3, 185–205. MR**1200422**, DOI 10.1016/0166-8641(92)90141-L - Helmut Hofer and Eduard Zehnder,
*Symplectic invariants and Hamiltonian dynamics*, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR**1306732**, DOI 10.1007/978-3-0348-8540-9 - Steven Hurder and Yoshihiko Mitsumatsu,
*Transverse Euler classes of foliations on nonatomic foliation cycles*, Differential topology, foliations, and group actions (Rio de Janeiro, 1992) Contemp. Math., vol. 161, Amer. Math. Soc., Providence, RI, 1994, pp. 29–39. MR**1271826**, DOI 10.1090/conm/161/01491 - A. B. Krygin,
*Continuation of diffeomorphisms retaining volume*, Funct. Anal. and Appl.**5**(1971), 147–150. - Dusa McDuff,
*Local homology of groups of volume preserving diffeomorphisms. I*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 4, 609–648 (1983). MR**707329** - John Milnor,
*On the existence of a connection with curvature zero*, Comment. Math. Helv.**32**(1958), 215–223. MR**95518**, DOI 10.1007/BF02564579 - Yoshihiko Mitsumatsu,
*On the self-intersections of foliation cycles*, Trans. Amer. Math. Soc.**334**(1992), no. 2, 851–860. MR**1183731**, DOI 10.1090/S0002-9947-1992-1183731-8 - Jürgen Moser,
*On the volume elements on a manifold*, Trans. Amer. Math. Soc.**120**(1965), 286–294. MR**182927**, DOI 10.1090/S0002-9947-1965-0182927-5 - Sergio Sispanov,
*Generalización del teorema de Laguerre*, Bol. Mat.**12**(1939), 113–117 (Spanish). MR**3** - Stephen Smale,
*Diffeomorphisms of the $2$-sphere*, Proc. Amer. Math. Soc.**10**(1959), 621–626. MR**112149**, DOI 10.1090/S0002-9939-1959-0112149-8 - John W. Wood,
*Bundles with totally disconnected structure group*, Comment. Math. Helv.**46**(1971), 257–273. MR**293655**, DOI 10.1007/BF02566843

## Additional Information

**Takashi Tsuboi**- Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro, Tokyo 153, Japan
- Email: tsuboi@ms.u-tokyo.ac.jp
- Received by editor(s): June 6, 1997
- Received by editor(s) in revised form: September 12, 1997
- Published electronically: March 18, 1999
- Additional Notes: The author was supported in part by Grant-in-Aid for Scientific Research 07454013 and 09440028, Ministry of Education, Science, Sports and Culture, Japan.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 515-524 - MSC (1991): Primary 57R32, 53C15; Secondary 57R50, 58H10, 53C12
- DOI: https://doi.org/10.1090/S0002-9947-99-02253-9
- MathSciNet review: 1487633